Volumetric imaging of holographic optical traps

ABSTRACT

A method and system for manipulating object using a three dimensional optical trap configuration. By use of selected hologram on optical strap can be configured as a preselected three dimensional configuration for a variety of complex uses. The system can include various optical train components, such as partially transmissive mirrors and Keplerian telescope components to provide advantageously three dimensional optical traps.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/974,716, filed Oct. 16, 2007, which claims priority from U.S.Provisional Application 60/852,252, filed Oct. 17, 2006, both of whichare incorporated herein by reference in their entirety.

This invention is directed toward volumetric imaging of holographicoptical traps. More particularly, the invention is directed to a methodand system for creating arbitrary pro-selected three-dimensional (3D)configurations of optical traps having individually specified opticalcharacteristics. Holographic techniques are used to modify individualtrap wavefronts to establish pre-selected 3D structures havingpredetermined properties and are positionable independently in threedimensional space to carry out a variety of commercially useful tasks.

The United States Government has certain rights in this inventionpursuant to a grant from the National Science Foundation through grantnumber DMR-0451589.

BACKGROUND OF THE INVENTION

There is, a well developed technology of using single light beams toform an optical trap which applies optical forces from the focused beamflight to confine an object to a particular location in space. Theseoptical traps, or optical tweezers, have enabled fine scale manipulationof objects for a variety of commercial purposes. In addition, linetraps, or extended optical tweezers, have been created which act as aone dimensional potential energy landscape for manipulating mesoscopicobjects. Such line traps can be used to rapidly screen interactionsbetween colloidal aid biological particles which find uses in biologicalresearch, medical diagnostics and drug discovery. However, theseapplications require methods of manipulation diagnostics and drugdiscovery. However, these applications require methods of manipulationfor projecting line traps with precisely defined characteristics whichprevent their use in situations with high performance demands. Further,the low degrees of freedom and facility of use for such line trapsreduces the ease of use and limits the types of uses available.

SUMMARY OF THE INVENTION

The facility and range of applications of optical traps is greatlyexpanded by the method and system of the invention in which 3D intensitydistributions are created by holography. These 3D representations arecreated by holographically translating optical traps through an opticaltrain's focal plane and acquiring a stack of two dimensional images inthe process. Shape phase holography is used to create a pre-selected 3Dintensity distribution which has substantial degrees of freedom tomanipulate any variety of object or mass for any task.

Various aspects of the invention are described hereinafter; and theseand other improvements are described in greater detail below, includingthe drawings described in the following section.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an optical train for performing a method of theinvention;

FIG. 2A illustrates a particular optical condition with z<0 for anobjective lens in the system of FIG. 1; FIG. 2B illustrates the opticalcondition for z=0 for the objective lens of FIG. 1 and FIG. 2Cillustrates the optical condition for z>0 for the objective lens of FIG.1;

FIG. 3A illustrates a 3D reconstruction of an optical tweezerpropagating along the z axis; FIG. 3B illustrates a cross-section ofFIG. 3A along an xy plane; FIG. 3C illustrates a cross-section of FIG.3A along a yz plane; FIG. 3D illustrates a cross-section of FIG. 3Aalong an xz plane; FIG. 3E illustrates a volumetric reconstruction of 35optical tweezers arranged in a body-centered cubic lattice of the typeshown in FIG. 3F;

FIG. 4A illustrates a 3D reconstruction of a cylindrical lens lineoptical tweezer; FIG. 4B illustrates a cross-section of FIG. 4A along anxy plane; FIG. 4C illustrates a cross-section of FIG. 4A along a yzplane; and FIG. 4D illustrates a cross-section of FIG. 4A along an xzplane; and

FIG. 5A illustrates a 3D reconstruction of a holographic optical trapfeaturing diffraction-limited convergence to a single focal plane; FIG.5B illustrates a cross-section of FIG. 5A along a xy plane; FIG. 5Cillustrates a cross-section of FIG. 5A along a yz plane; and FIG. 5Dillustrates a cross-section of FIG. 5A along an xz plane.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

An optical system for performing, methods of the invention isillustrated generally at 10 in FIG. 1. A beam of light 20 is output froma frequency-doubled solid-state laser 30, preferably a Coherent Verdisystem operating at a wavelength of λ=532 nm. The beam of light 20 isdirected to an input pupil 40 of a high-numerical-aperture objectivelens 50, preferably a Nikon 100×Plan Apo, NA 1.4, oil immersion systemthat focuses the beam of light 20 into an optical trap (not shown). Thebeam of light 20 is imprinted with a phase-only hologram by acomputer-addressed liquid-crystal spatial light modulator 60 (“SLM 60”),preferably a Hamamatsu X8267 PPM disposed in a plane conjugate to theobjective lens' 50 input plane. Computer 95 executes conventionalcomputer software to generate the appropriate hologram using the SLM 60.As a result, the light field, ψ(r), in the objective lens' 50 focalplane is related to the field ψ(ρ) in the plane of the SLM 60 by theFraunhofer transform,

$\begin{matrix}{{{\psi(r)} = {{- \frac{\mathbb{i}}{\lambda\; f}}{\int_{\Omega}^{\;}{{\psi(\rho)}\ {\exp( {{- {\mathbb{i}}}\frac{2\pi}{\lambda\; f}{r \cdot \rho}} )}{\mathbb{d}^{2}\rho}}}}},} & (1)\end{matrix}$where f is the objective's focal length, where Ω is the optical train'saperture, and where we have dropped irrelevant phase factors. Assumingthat the beam of light 20 illuminates the SLM 60 with a radiallysymmetric amplitude profile, u(ρ), and uniform phase, the field in theSLM's plane may be written as,ψ(ρ)=u(ρ)exp(iφ(ρ)),  (2)where φ(ρ) is the real-valued phase profile imprinted on the beam oflight 20 by the SUM 60. The SLM 60 in our preferred form of the system10 imposes phase shifts between 0 and 2 π radians at each pixel of a768×768 array. This two-dimensional phase array can be used to project acomputer-generated phase-only hologram, φ(ρ), designed to transform thesingle optical tweezer into any desired three-dimensional configurationof optical traps, each with individually specified intensities andwavefront properties.

Ordinarily, the pattern of holographic optical traps would be put to useby projecting it into a fluid-borne sample mounted in the objectivelens' 50 focal plane. To characterize the light field, we instead mounta front-surface mirror 70 in the sample plane. This mirror 70 reflectsthe trapping light back into the objective lens 50, which transmitsimages of the traps through the partially reflecting mirror 70 to acharge-coupled device (CCD) camera 80, preferably a NEC TI-324AII. Inour implementation, the objective lens 50, the camera 80 and cameraeyepiece (not shown), are mounted in a conventional optical Microscope(not shown) and which is preferably a Nikon TE-2000U.

Three-dimensional reconstructions of the optical traps' intensitydistribution can be obtained by translating the mirror 70 relative tothe objective lens 50. Equivalently, the traps can be translatedrelative to the mirror 70 by superimposing the parabolic phase function,

$\begin{matrix}{{{\varphi_{z}(\rho)} = {- \frac{{\pi\rho}^{2}z}{\lambda\; f^{2}}}},} & (3)\end{matrix}$onto the hologram φ₀(ρ) encoding a particular pattern of traps. Thecombined hologram, φ₀(ρ)=φ₀(ρ)+φ_(z)(ρ) mod 2 π, projects the samepattern of traps as φ₀(ρ) but with each trap translated by−z alongoptical axis 90 of the system 10. The resulting image obtained from thereflected light represents a cross-section of the original trappingintensity at distance z from the focal plane of the objective lens 50.Translating the traps under software control by computer 95 isparticularly convenient because it minimizes changes in the opticaltrain's properties due to mechanical motion and facilitates moreaccurate displacements along the optical axis 90. Images obtained ateach value of z are stacked up to yield a complete volumetricrepresentation of the intensity distribution.

As shown schematically in FIGS. 2A-2C, the objective lens 50 capturesessentially all of the reflected light for z<0. For z>0, however, theoutermost rays of the converging trap are cut off by the objective lens'50 output pupil 105, and the contrast is reduced accordingly. This couldbe corrected by multiplying the measured intensity field by a factorproportional to z for z>0. The appropriate factor, however, is difficultto determine accurately, so we present only unaltered results.

FIG. 3A shows a conventional optical tweezer 100 reconstructed in themanner described hereinbefore and displayed as an isointensity surfaceat 5 percent peak intensity and in three cross-sections (FIGS. 3B-3D).The representation in FIG. 3A is useful for showing the overallstructure of the converging light, and the cross-sections of FIGS. 3B-3Dprovide an impression of the three dimensional light field that willconfine an optically trapped object. The angle of convergence of 63° inimmersion oil obtained from these data is consistent with an overallnumerical aperture of 1.4. The radius of sharpest focus, r_(min)≈0.2 μm,is consistent with diffraction-limited focusing of the beam of light 20.

These results highlight two additional aspects of this reconstructiontechnique. The objective lens 50 is designed to correct for sphericalaberration when the beam of light 20 passing through water is refractedby a glass coverslip. Without this additional refraction, the projectedoptical trap 100 actually is degraded by roughly 20λ of sphericalaberration, introduced by the objective lens 50. This reduces theapparent numerical aperture and also extends the trap's focus along thez axis. The trap's effective numerical aperture in water would beroughly 1.2. The effect of spherical aberration can be approximatelycorrected by pre-distorting the beam of light 20 with the additionalphase profile,

$\begin{matrix}{{{\varphi_{a}(\rho)} = {\frac{a}{\sqrt{2}}( {{6x^{4}} - {6x^{2}} + 1} )}},} & (4)\end{matrix}$the Zernike polynomial describing spherical aberration. The radius, x,is measured as a fraction of the optical train aperture, and thecoefficient α is measured in wavelengths of light. This procedure isused to correct for small amount of aberration present in practicaloptical trapping systems to optimize their performance.

This correction was applied to an array 110 of 35 optical tweezers shownas a three-dimensional reconstruction in FIG. 3E. These optical traps100 are arranged in a three-dimensional body-centered cubic (BCC)lattice 115 shown in FIG. 3F with a 10.8 μm lattice constant. Withoutcorrecting for spherical aberration, these traps 100 would blend intoeach other along the optical axis 90. With correction, their axialintensity gradients are clearly resolved. This accounts for holographictraps' ability to organize objects along the optical axis.

Correcting for aberrations reduces the range of displacements, z, thatcan be imaged. Combining φ_(a) (ρ) with φ_(z)(ρ) and φ₀(ρ) increasesgradients in φ(ρ), particularly for larger values of ρ near the edges ofthe diffraction optical element. Diffraction efficiency falls offrapidly when |∇φ(ρ)| exceeds 2π/Δρ, the maximum phase gradient that canbe encoded on the SLM 60 with pixel size Δρ. This problem is exacerbatedwhen φ₀(ρ) itself has large gradients. In a preferred embodiment morecomplex trapping patterns without aberration are prepared. Inparticular, we use uncorrected volumetric imaging to illustrate thecomparative advantages of the extended optical traps 100.

The extended optical traps 100 have been projected in a time-sharedsense by rapidly scanning a conventional optical tweezer along thetrap's intended contour. A scanned trap has optical characteristics asgood as a point-like optical tweezer, and an effective potential energywell that can be tailored by adjusting the instantaneous scanning rateKinematic effects due to the trap's motion can be minimized by scanningrapidly enough. For some applications, however, continuous illuminationor the simplicity of an optical train with no scanning capabilities canbe desirable.

Continuously illuminated line traps have been created by expanding anoptical tweezer 125 along one direction (see FIG. 4A). This can beachieved, for example, by introducing a cylindrical lens component suchas by element 130 (see FIG. 1) into the objective's input plane.Equivalently, a cylindrical-lens line tweezer can be implemented byencoding the function φ_(c)(ρ)=πz₀ρ_(x) ²/(λƒ²) on the SLM 60. Theresult, shown in FIGS. 4A-4D appears best useful in the plane of bestfocus, z=z₀, with the point-like tweezer having been extended to a linewith nearly parabolic intensity and a nearly Gaussian phase profile. Thethree-dimensional reconstruction, however, reveals that the cylindricallens component merely introduces a large amount of astigmatism into thebeam of light 20, creating a second focal line perpendicular to thefirst. This is problematic for some applications because the astigmaticbeam's axial intensity gradients are far weaker than a conventionaloptical tweezer's. Consequently, cylindrical-lens line traps typicallycannot localize objects against radiation pressure along the opticalaxis 90.

Replacing the single cylindrical lens with a cylindrical Kepleriantelescope for the element 130 eliminates the astigmatism and thuscreates a stable three-dimensional optical trap. Similarly, using theobjective lens 50 to focus two interfering beams creates aninterferometric optical trap capable of three-dimensional trapping.These approaches, however, offer little control over the extended trapsintensity profiles, and neither affords control over the phase profile.

Shape-phase holography provides absolute control over both the amplitudeand phase profiles of an extended form of the optical trap 100 at theexpense of diffraction efficiency. It also yields traps with optimizedaxial intensity gradients, suitable for three-dimensional trapping. Ifthe line trap is characterized by an amplitude profile ũ(ρ_(x)) and aphase profile {tilde over (p)}(ρ_(x)) along the {circumflex over(ρ)}_(x) direction in the objective's focal plane, then the field in theSLM plane is given from Eq. (1) as,ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))),  (5)where the phase p(ρ_(x)) is adjusted so that u(ρ_(x))≧0. Shape-phaseholography implements this one-dimensional complex wavefront profile asa two-dimensional phase-only hologram,

$\begin{matrix}{{\varphi(\rho)} = \{ \begin{matrix}{{p( \rho_{x} )},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{{S(\rho)} = 0},}\end{matrix} } & (6)\end{matrix}$where the shape function S(ρ) allocates a number of pixels along the rowρ_(y) proportional to u(ρ_(x)). One particularly effective choice is forS(ρ) to select pixels randomly along each row in the appropriaterelative numbers. The unassigned pixels then are given values q(ρ) thatredirect the excess light away from the intended line. Typical resultsare presented in FIG. 5A.

Unlike the cylindrical-lens trap, the holographic line trap 130 in FIGS.5A-5D focuses as a conical wedge to a single diffraction-limited line inthe objective's focal plane. Consequently, its transverse angle ofconvergence is comparable to that of an optimized point trap. This meansthat the holographic line trap 120 has comparably strong axial intensitygradients, which explains its ability to trap objects stably againstradiation pressure in the z direction.

The line trap's transverse convergence does not depend strongly on thechoice of intensity profile along the line. Its three-dimensionalintensity distribution, however, is very sensitive to the phase profilealong the line. Abrupt phase changes cause intensity fluctuationsthrough Gibbs phenomenon. Smoother variations do not affect theintensity profile along the line, but can substantially restructure thebeam. The line trap 120 created by the cylindrical lens element 130 forexample, has a parabolic phase profile. Inserting this choice into Eq.(2) and calculating the associated shape-phase hologram with Eqs. (1)and (6) yields the same cylindrical lens phase profile. This observationopens the door to applications in which the phase profile along a linecan be tuned to create a desired three-dimensional intensitydistribution, or in which the measured three-dimensional intensitydistribution can be used to assess the phase profile along the line.These applications will be discussed elsewhere.

The foregoing description of embodiments of the present invention havebeen presented for purposes of illustration and description. It is notintended to be exhaustive or to limit the present invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of thepresent invention. The embodiments were chosen and described in order toexplain the principles of the present invention and its practicalapplication to enable one skilled in the art to utilize the presentinvention in various embodiments, and with various modifications, as aresuited to the particular use contemplated.

1. A system for obtaining a three-dimensional cross-section of thetrapping intensity of trapping light for at least one of characterizingthe trapping light and for performing tasks on a sample, comprising: anoptical train; a source for providing a beam of light to the opticaltrain; a computer for applying a predetermined hologram to the beam oflight to generate trapping light; a mirror for reflecting the trappinglight; a sensor for obtaining a plurality of three-dimensionalcross-sections of trapping intensity of the trapping light from thereflected light thereby enabling at least one of analysis of thetrapping intensity and use of the trapping intensity cross-section forguiding use of the trapping light on the sample.
 2. The system asdefined in claim 1 further including a front surface form of the mirrorwith the trapping light reflected off the front surface mirror into anobjective lens and further transmits images of the trapping lightthrough a partially reflecting mirror into a camera.
 3. The system asdefined in claim 1 wherein the trapping light is translated relative tothe mirror by the computer superimposing a phase function on thetrapping light.
 4. The system as defined in claim 1 wherein an image isconstructed from the reflected light which represents a cross-section ofthe trapping intensity reconstructed at least one of (1) a plurality ofdistances z from a focal plane of the optical train; and (2) with aplurality of the images being used to create a volumetric representationof the trapping intensity.
 5. The system as defined in claim 1 whereinthe computer determines a hologram to carry out at least one of theanalysis of the trapping light or use of the cross-section for operatingon the sample and further includes having the optical train with amirror disposed in the sample plane and operating on the beam of lightto process a light field of the trapping light.
 6. The system as definedin claim 5 wherein the hologram includes a parabolic phase function toenable translating the trapping light, the system configured to create aplurality of cross-sections of trapping intensity by at least one of (1)the mirror disposed in the sample plane translating relative to anobjective lens of the optical trapping and (2) the trapping lighttranslating relative to a fixed form of the mirror.
 7. The system asdefined in claim 6 wherein the parabolic phase function comprises,${{\varphi_{z}(\rho)} = {- \frac{{\pi\rho}^{2}z}{\lambda\; f^{2}}}},$where, φ=hologram phase value at position ρ; z=distance along an opticalaxis of the optical train; λ=wavelength of the light; ƒ=objective lensfocal length; ρ=position in the hologram.
 8. The system as defined inclaim 6 wherein a computer program executed by the computer determinesthe hologram for translating the trapping light.
 9. The system asdefined in claim 5 wherein the phase only hologram comprises,${\varphi(\rho)} = \{ \begin{matrix}{{p( \rho_{x} )},} & {{S(\rho)} = 1} \\{{q(\rho)},} & {{{S(\rho)} = 0},}\end{matrix} $ where S (ρ) is a shape function which allocates anumber of pixels along row, ρ; p (ρ) is a phase profile and q (ρ) arevalues along the row, ρ, for unassigned pixels along the row and whichredirect excess light from an intended line.
 10. The system as definedin claim 5 wherein the sensor provides analysis data defining a twodimensional cross section of original trapping density of thepredetermined trapping light configuration.
 11. The system as defined inclaim 10 wherein the data includes a plurality of the two dimensionalcross sections.
 12. The system as defined in claim 1 wherein thetrapping light is configured to manipulate an object in the same plane.13. The system as defined in claim 12 further including the step ofpassing the beam of light through an objective lens of the optical trainand applying a phase profile to the trapping light, thereby enabling aplurality of trapping lights to organize selected ones of the objectsalong an optical axis of the optical train.
 14. The system as defined inclaim 13 wherein the phase profile comprises,${{\varphi_{a}(\rho)} = {\frac{a}{\sqrt{2}}( {{6x^{4}} - {6x^{2}} + 1} )}},$thereby correcting for spherical aberration where x is distance measuredas a fraction of the optical train aperture and “a” is a coefficientmeasured in wavelengths of light.
 15. The system as defined in claim 12further including the step of forming a single trapping light andrapidly scanning the single trapping light along a predeterminedconfiguration to enable manipulation of the sample.
 16. The system asdefined in claim 1 further including an objective lens disposed in theoptical train and further includes a cylindrical lens component disposedin an input plane of the objective lens.
 17. The system as defined inclaim 1 wherein the predetermined hologram includes a cylindrical-lensline component.
 18. The system as defined in claim 17 wherein thepredetermined hologram includes the function φ_(c)(ρ)=πz₀ρ_(x) ²/(λƒ²).19. The system as defined in claim 1 further including in the opticaltrain a Keplerian telescope for eliminating astigmatism, therebycreating a stable form of three dimensional trapping light.
 20. Thesystem as defined in claim 19 wherein the optical train includes an SLMhaving an associated plane and the field in the associated plane isgiven by, ψ(ρ)=u(ρ_(x))exp(ip(ρ_(x))).
 21. The system as defined inclaim 1 wherein the trapping light comprises a hologram constructed of,a. an amplitude profile ũ(ρ_(x)), b. a phase profile {tilde over(p)}(ρ_(x)) with an objective focal plane direction {circumflex over(ρ)}_(x).